Two carts with masses of 13.2 kg and 3.0 kg

move in opposite directions on a frictionless
horizontal track with speeds of 6.8 m/s and
3.5 m/s, respectively. The carts stick together
after colliding head-on.
Find their final speed.
Answer in units of m/s.

M1*V1 + M2*V2 = M1*V + M2*V.

13.2*6.8 + 3*(-3.5) = 13.2V + 3V.
V = ?

To find the final speed of the carts after they stick together, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.

The formula for momentum is:

momentum = mass × velocity

Let's represent the mass and velocity of the first cart as m1 and v1, respectively, and the mass and velocity of the second cart as m2 and v2, respectively.

The total momentum before the collision is given by:

initial momentum = (m1 × v1) + (m2 × v2)

After the collision, the two carts stick together and move with a common velocity, let's call it vf.

The total momentum after the collision is simply the mass of the combined carts multiplied by the final velocity:

final momentum = (m1 + m2) × vf

According to the conservation of momentum principle, the initial momentum must be equal to the final momentum, so:

(m1 × v1) + (m2 × v2) = (m1 + m2) × vf

Now, plug in the given values:

m1 = 13.2 kg
v1 = 6.8 m/s
m2 = 3.0 kg
v2 = 3.5 m/s

(13.2 kg × 6.8 m/s) + (3.0 kg × 3.5 m/s) = (13.2 kg + 3.0 kg) × vf

89.76 kg*m/s + 10.5 kg*m/s = 16.2 kg × vf

100.26 kg*m/s = 16.2 kg × vf

Divide both sides of the equation by 16.2 kg:

(100.26 kg*m/s) / (16.2 kg) = vf

vf = 6.1820 m/s

Therefore, the final speed of the combined carts is approximately 6.1820 m/s.

To find the final speed of the carts after they stick together, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.

The momentum of an object is given by the product of its mass and velocity:
momentum = mass × velocity.

Before the collision, the momentum of the first cart is:
momentum1 = mass1 × velocity1 = 13.2 kg × 6.8 m/s.

Before the collision, the momentum of the second cart is:
momentum2 = mass2 × velocity2 = 3.0 kg × (-3.5 m/s) [since it moves in the opposite direction].

The negative sign is used for the second cart's velocity because it's moving in the opposite direction.

Since momentum is conserved, the total momentum after the collision is equal to the sum of the initial momenta:
total momentum = momentum1 + momentum2.

After the carts stick together, they become one system. The mass of the combined system is the sum of the masses of the two carts:
mass of combined system = mass1 + mass2.

The final speed of the combined system can be found using the equation:
final speed = total momentum / (mass of combined system).

Plugging in the values we have:
total momentum = momentum1 + momentum2 = 13.2 kg × 6.8 m/s + 3.0 kg × (-3.5 m/s).
mass of combined system = mass1 + mass2 = 13.2 kg + 3.0 kg.

Calculating these, we find:
total momentum = 89.76 kg·m/s - 10.5 kg·m/s = 79.26 kg·m/s.
mass of combined system = 13.2 kg + 3.0 kg = 16.2 kg.

Finally, dividing the total momentum by the mass of the combined system gives us the final speed:
final speed = 79.26 kg·m/s / 16.2 kg = 4.89 m/s.

Therefore, the final speed of the carts after colliding head-on and sticking together is 4.89 m/s.